2. Geometric deep learning foundations

Crash deformation fields are not defined on a regular pixel grid. A vehicle body after impact is a curved, torn, folded shell whose material points have moved, rotated, and stretched. To learn a mapping from pre-crash geometry to post-crash displacement, we must first decide what mathematical object carries the signal. Geometric deep learning offers four canonical domains — graphs, meshes, point clouds, and manifolds — and each imposes a different inductive bias on how deformation is represented, aggregated, and predicted. Choosing the wrong domain is not a cosmetic decision; it determines which physical invariances the network can respect and which artifacts it will hallucinate.

Why the domain choice matters for deformation

A deformation field is a vector-valued function u: M → ℝ³ assigning each material point a displacement. The domain M is the reference (undeformed) configuration. Three properties of M drive the architecture: its sampling (are points ordered, connected, or unstructured?), its connectivity (do we know which points are neighbors?), and its metric (can we measure geodesic distance along the surface?). Graphs give connectivity without metric; point clouds give neither; meshes give both discretely; manifolds give both continuously. Crash structures — stamped steel panels, extruded rails, welded joints — are physically continuous shells, so the manifold is the ground truth and the other three are discretizations of it.

Graphs: connectivity-first deformation propagation

A graph G = (V, E) with node features h_v and edge features e_uv is the natural abstraction when the structure is a skeleton rather than a surface: a body-in-white frame modeled as beam elements, a crash box with spot-weld edges, or a coarse connectivity graph extracted from a finite element model. Message passing updates each node by aggregating neighbor messages:

m_v = AGG({ φ(h_u, h_v, e_uv) : u ∈ N(v) })
h_v' = ψ(h_v, m_v)

For deformation learning, node features typically encode reference position, local surface normal, and material thickness; edge features encode rest length and stiffness class. The key limitation: a graph has no notion of face orientation or surface area, so it cannot directly enforce shell bending energy. Graphs are best used for global load-path reasoning — predicting which structural members absorb energy — and for coarse-to-fine initialization before a mesh-level refinement.

Meshes: the workhorse for shell deformation

A triangle mesh (V, F) adds faces, and with faces come normals, areas, and discrete differential operators. This is decisive for crash deformation because the physics is dominated by bending and stretching of thin shells. The cotangent Laplacian

L_ij = (cot α_ij + cot β_ij) / 2   for edge (i,j)
(L u)_i = Σ_j L_ij (u_i − u_j)

approximates the Laplace–Beltrami operator and lets a network penalize non-smooth displacement fields or solve for harmonic interpolation of sparse measurements. Mesh convolutional and graph-attention layers operating on (V, F) can condition on face normals and dihedral angles, which is exactly the information needed to distinguish a sharp fold (high curvature, plastic hinge) from a smooth bulge (elastic membrane stretching). In practice, a mesh-based deformation network takes the undeformed vertices X ∈ ℝ^{N×3}, a conditioning vector (impact velocity, barrier offset, material grade), and outputs U ∈ ℝ^{N×3}; the loss combines data fidelity at sensor locations with a bending-energy regularizer ‖L U‖².

Point clouds: measurement-native but connectivity-poor

Laser scans and photogrammetry of a crashed vehicle yield unordered point sets P = {p_i ∈ ℝ³} with no explicit topology. Point-cloud networks (PointNet-style symmetric aggregation, or continuous convolutions) are permutation-invariant and handle irregular density, which matches scan reality. However, without connectivity, the network must infer neighborhoods via k-NN or radius search at every layer, and it cannot natively compute face normals or discrete curvature. For deformation learning, point clouds are strongest as the observation domain — registering a post-crash scan to the CAD reference and sampling displacement constraints — and weaker as the prediction domain, where missing connectivity causes over-smoothing across folds.

Manifolds: the continuous ideal and its discretization

The manifold view treats the body panel as a smooth 2-manifold M embedded in ℝ³, with a Riemannian metric inherited from the embedding. This is the correct physical setting: strain is a metric tensor change, and bending is a change in second fundamental form. In practice we never compute on the continuous manifold; we discretize it as a mesh and use the discrete operators above. The manifold framing matters because it tells us which invariances are physical: the deformation field should be equivariant to rigid-body motion of the whole vehicle (translation and rotation), and invariant to the choice of parameterization. Architectures that respect SE(3) equivariance — using invariant scalar features and equivariant vector features — generalize far better across impact orientations than plain coordinate MLPs.

Choosing and combining domains in practice

The four domains are not mutually exclusive; a robust crash-deformation learner usually composes them. A concrete, reproducible example: take a simplified front-rail assembly discretized as a triangle mesh with N ≈ 12,000 vertices. Build a mesh graph-attention encoder with 4 layers, hidden width 128, using vertex features [x, y, z, n_x, n_y, n_z, thickness] and edge features [rest_length, dihedral_angle]. Condition on a 3-vector [impact_speed, barrier_overlap, material_yield] injected via FiLM at each layer. Output U ∈ ℝ^{N×3}. Train with L = L_data + λ_bend · ‖L U‖², where L_data is masked MSE at the ~200 vertices corresponding to available scan or simulation samples and λ_bend = 0.05.

Verification is domain-specific. Check (1) rigid-motion equivariance: rotate the input mesh by a random R ∈ SO(3) and confirm the predicted displacement rotates by the same R within numerical tolerance; a large deviation indicates coordinate leakage. (2) Fold preservation: inspect the dihedral-angle distribution at predicted plastic hinges; over-smoothed predictions show a collapsed tail where sharp folds should be. (3) Registration residual: align the predicted post-crash mesh to a held-out scan with ICP and report mean point-to-plane distance; values above roughly 8–10 mm on a passenger-car front end typically signal under-fitting at the crush zone. Typical failure symptoms map cleanly to domain mistakes: excessive smoothing across folds points to a point-cloud-style symmetric aggregator used on a mesh; spurious high-frequency ripples point to a missing bending regularizer; orientation-dependent predictions point to non-equivariant coordinate inputs.

The practical rule is to let the physics pick the domain. Energy absorption along structural members → graph. Bending and stretching of panels → mesh. Measurement and registration → point cloud. Invariance guarantees and continuous strain definitions → manifold. A deformation learner that respects this mapping will generalize across impact configurations far better than one that treats geometry as an undifferentiated coordinate list.